ON A TWO-VARIABLES FRACTIONAL PARTIAL DIFFERENTIAL INCLUSION VIA RIEMANN-LIOUVILLE DERIVATIVE

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1 Novi Sad J. Math. Vol. 46, No. 2, 26, ON A TWO-VARIABLES FRACTIONAL PARTIAL DIFFERENTIAL INCLUSION VIA RIEMANN-LIOUVILLE DERIVATIVE S. Etemad and Sh. Rezapour 23 Abstract. We investigate the existence a solution to a two-variable fractional partial differential inclusion via Riemann-Liouville derivative. Also, we provide an example to illustrate our main result. AMS Mathematics Subject Classification 2: 26A33; 34A; 34A8 Key words and phrases: boundary value problem; endpoint; fixed point; fractional partial derivative; two-variables fractional partial differential inclusion. Introduction There are many published works about fractional partial differential equations by using the notions of delay or time-fractional see for example,, 2, 9 and. It is interesting to work on two variables fractional partial differential equations see, for example, 3, 4, 6 and. Let, and α α, α 2 where < α, α 2. Also, put J a J b, a, b where a and b are positive constants. The Riemann-Liouville fractional partial integral of u L J a J b is defined by I α ux, y x s α y t α 2 us, tdtds whenever the integral exists see, for example, 4, 5 and 6. The Riemann- Liouville partial derivative of fractional order α for a function u L J a J b is defined by D α ux, y D 2 xyi α ux, y 2 x y see for more details 4, 5 and 6. Note that whenever β β, β 2 > 6. I α I β ux, y Iαβ ux, y x s α y t α 2 Γ α Γ α 2 dtds Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran sina.etemad@gmail.com 2 Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Iran sh.rezapour@azaruniv.edu 3 Corresponding author

2 46 S. Etemad, Sh. Rezapour Let X, d be a metric space, PX the class of all nonempty subsets of X, P cl X the class of all closed subsets of X, P bd X the class of all bounded subsets of X, P cp X the class of all compact subsets of X and P cv X the class of all convex subsets of X. A multi-valued map F : J a J b P cl R is measurable whenever the function x, y dw, F x, y inf{ w v : v F x, y} is measurable for all w R, where J a J b, a, b 2. Also, the Pompeiu-Hausdorff metric H d : PX PX, is defined by H d A, B max{sup da, B, sup da, b}, a A b B where da, b inf a A da, b 7. Then P cl,bd X, H d is a metric space and P cl X, H d is a generalized metric space 7. Recall that a multifunction F : X PX is said to be a contraction if there exists k, such that H d F u, F v kdu, v for all u, v X 3. An element u X is called endpoint of the multifunction F : X PX whenever F u {u} 8. We say that the multifunction F has an approximate endpoint property whenever inf u X sup w F u du, w 8. A real-valued function f on R is called upper semi-continuous whenever lim sup n fλ n fλ for all sequence {λ n } n with λ n λ. In this paper, using the main idea of 5, 6 and 7, we investigate the existence of solutions for the two-variables fractional partial differential inclusion. D α ux, y F x, y, ux, y, with the partial integral boundary value conditions.2 I α ux, λ φx, I α u, y λ 2 γy, where D α denotes the Riemann-Liouville fractional partial derivative of order α, x, y J a J b, < α i, λ i R i, 2 and F : J a J b R PR is a compact valued multi-valued map. Here, the functions φ : J a R and γ : J b R are absolutely continuous with φ γ. We need the following endpoint result. Theorem.. 8 Suppose that X, d is a complete metric space, ψ :,, is an upper semi-continuous function such that ψt < t and lim inf t t ψt > for all t > and T : X CBX is a multifunction such that H d T x, T y ψdx, y for all x, y X. Then T has a unique endpoint if and only if T has approximate endpoint property. 2. Main results Now we are ready to state and prove our main results. First, we give the following key result. Lemma 2.. Let f LJ a J b and α α, α 2,,. Then the continuous function u LJ a J b is a solution for the fractional partial differential equation 2. D α ux, y fx, y

3 On a two-variables fractional partial differential inclusion with boundary conditions I α ux, λ φx and I α u, y λ 2 γy if and only if u is a solution for the fractional integral equation ux, y λ 2x α Γα Iα γy λ y α2 Γα 2 Iα φx I α fx, y. Proof. Let u be a solution for the fractional partial differential equation 2.. Then, we have DxyI 2 α u x, y fx, y and so I α u x, y I α u x, I α u, y I α u, I fx, y. Using the boundary conditions we get I α u x, y λ φx λ 2 γy λ φ I fx, y and so I α u x, y I fx, y λ φx λ 2 γy. Since I u x, y u x, y, we obtain I α u x, y I x, αfx, y y λ φxλ 2 γy. On the other hand, we have I α I α u x, y I α fx, yx, y x, y I α λ φx λ 2 γy and so I u x, y I α fx, y x, y I α But we have I α px, y Since φx I α px, y λ λ 2 φsds and γy λ λ 2 λ y α2 Γα Γα 2 λ φx λ 2 γy I α px, y. 2.3 x s α y t α 2 φsdtds x s α y t α 2 γtdtds. γtdt, we obtain x s α y t α 2 s x s α y t α 2 t x s α s t φτdτds λ 2 x α y t α2 γτdτdt Γα Γα 2 λ y α 2 x s s τ α φτdτ Γα 2 Γα λ 2 x α y t t τ α 2 γτdτ Γα Γα 2 λ y α 2 Γα 2 I α φsds λ 2x α Γα ds dt φτdτ dtds γτdτ dtds I α2 γtdt.

4 48 S. Etemad, Sh. Rezapour Since I α and φx LJ a and I α2 γy LJ b, the functions I α φsds I α2 γtdt are absolutely continuous and so there exists DxyI 2 α px, y for almost all x, y J a J b. By applying the operator Dxy 2 on both sides of 2.3, we get Dxy 2 I u x, y I αfx, y x, y Dxy I 2 αpx, y. Thus, Dxy 2 λ y α2 Γα 2 D x λ α 2 y α 2 Γα 2 u x, y I α fx, y I α φsds λ 2x α Γα I α φsds λ 2x α Γα Iα2 λ y α2 Γα 2 Iα φx λ 2x α Γα Iα 2 γy. I α 2 γtdt γy Hence, u x, y λ y α 2 Γα 2 Iα φx λ 2x α Γα Iα 2 γy I α fx, y. This shows that u is a solution of the fractional integral equation 2.2. Now, let u be a solution for the fractional integral equation 2.2. Then, I α u x, y I α λ y α 2 Γα 2 Iα hand by using Bz, w λ 2 λ 2 λ 2 Γα Γ α φx λ 2x α Γα Iα 2 ψy x, yi fx, y. On the other x w x z dx ΓzΓw, we get Γz w I α λ2 x α Γα Iα2 γy x, y x s α y t α 2 s α Γ α Γ α 2 Γα Γ α Γ α 2 x s α s α ds Γα Iα 2 γt dtds x s α s α y t α 2 I α2 γtdtds Γ α 2 λ 2Bα, α I α 2 I α 2 Γα Γ α γt y λ 2 Γ I γy λ 2 γy γ λ 2 γy and similarly I α λ y α 2 Γα 2 Iα φx x, y λ φx. Thus, 2.3 I α u x, y λ 2 γy λ φx I fx, y. y t α2 I α 2 γtdt

5 On a two-variables fractional partial differential inclusion By applying the operator D 2 xy on both sides of 2.3, we obtain D 2 xy I α u x, y Dxy 2 λ 2 γy λ φx I fx, y and so D α u x, y fx, y. By using 2.3, we get I α u x, λ 2 γ λ φx I fx, λ φx and I α u, y λ 2 γy λ φ I f, y λ 2γy. This completes the proof. Consider the Banach space X CJ a J b, R endowed with the norm u sup x,y Ja J b ut. For u X, define the set of selections of F by S F,u : {v L J a J b, R : vx, y F x, y, ux, y for almost all x, y J a J b }. It has been proved that S F,u for all u CJ a J b, X. We say that u X is a solution for the boundary value problem.-.2 whenever it satisfies the boundary value conditions.2 and also there is a function v L J a J b, R such that vx, y F x, y, ux, y for all x, y J a J b and ux, y λ 2x α Γα Iα 2 γy λ y α2 Γα 2 Iα φx x s α y t α 2 vs, tdtds for almost all x, y J a J b. Define the multifunction N : X PX by N u {h X : hx, y λ 2x α Here, we provide our main result. Γα Iα 2 γy λ y α 2 Γα 2 Iα φx x s α y t α 2 vs, tdtds} for all x, y J a J b }. Theorem 2.2. Suppose that ψ :,, is a nondecreasing upper semi-continuous map such that lim inf t t ψt > and ψt < t for all t >, F : J a J b R P cp R is an integrable bounded multifunction such that F,, u : J a J b P cp R is measurable for all u R. Assume that there exits m CJ a J b,, such that H d F x, y, u F x, y, u Λ mx, yψ u u { for all x, y J a J b and u, u a α b α 2 } R, where Λ m. If Γα Γα 2 the multifunction N has the approximate endpoint property, then the fractional partial differential inclusion problem.-.2 has a solution.

6 5 S. Etemad, Sh. Rezapour Proof. First, we prove that the multifunction N has at least one endpoint. Let u X. Since the multivalued map x, y F x, y, ux, y is measurable and is closed-value, it has measurable selection and so S F,u is nonempty. Let {p n } n be a sequence in N u with p n p. For each n, choose v n S F,un such that p n x, y λ 2x α Γα Iα2 γy λ y α2 Γα 2 Iα φx x s α y t α 2 v n s, tdtds for all x, y J a J b. Since the operator F is compact, the sequence {v n } n has a subsequence converging to some v L J a J b. We denote this subsequence again by {v n } n. It is easy to see that v S F,u and px, y λ 2x α Γα Iα 2 γy λ y α2 Γα 2 Iα φx x s α y t α 2 vs, tdtds for all x, y J a J b. This shows that p N u and so N u is closed. Note that, N u is bounded because F has a compact values. Now, we show that H d N u, N w ψ u w for all u, w X. Let u, w X and h N w. Choose v S F,w such that h x, y λ 2x α Γα Iα 2 γy λ y α 2 Γα 2 Iα φx x s α y t α2 v s, tdtds for almost all x, y J a J b. By using the hypothesis, we have H d F x, y, ux, y F x, y, wx, y Λ mx, yψ ux, y wx, y and so we can choose z F x, y, ux, y such that v x, y z Λ mx, yψ ux, y wx, y. Define the multivalued map U : J a J b PR by Ux, y {z R : v x, y z Λ mx, yψ ux, y wx, y. Since v and η mψ u w are measurable, the multifunction U, Λ F,, u, is measurable. Hence, there exists v 2 x, y F x, y, ux, y such

7 On a two-variables fractional partial differential inclusion... 5 that v x, y v 2 x, y Λ mx, yψ ux, y wx, y. Now, consider the element h 2 N u defined by h 2 x, y λ 2x α Γα Iα 2 γy λ y α 2 Γα 2 Iα φx x s α y t α2 v 2 s, tdtds for all x, y J a J b. Put sup x,y Ja J b mx, y m. Then, we have h x, y h 2 x, y λ 2x α λ 2x α Γα Iα 2 Γα Iα 2 γy λ y α2 Γα 2 Iα φx x s α y t α 2 v s, tdtds γy λ y α 2 Γα 2 Iα φx x s α y t α2 v 2 s, tdtds x s α y t α2 v s, t v 2 s, t dtds { m ψ u w Λ x α y α 2 Γα Γα 2 Λ Λ ψ u w ψ u w and so h h 2 sup x,y Ja J b h x, y h 2 x, y ψ u w. Hence, H d N u, N w ψ u w for all u, w X. Since the multifunction N has approximate endpoint property according to Theorem., there exists u X such that N u {u }. It is easy to check that u is a solution for the fractional partial differential inclusion problem.-.2. For illustration of our main result, we give the following example. Example 2.3. Consider the fractional partial differential inclusion D α.3xy sin ux, y ux, y, sin ux, y with boundary value conditions I α ux,.e x and I α u, y.y 2, where x, y,,. Let α α, α 2 with α, α 2,, λ. and λ 2.. Define the multifunction F :,, R PR.3xy sin zt by F x, y, z,. If m :,,, is defined by sin zt }

8 52 S. Etemad, Sh. Rezapour mx, y 3 3 xy, then m. Consider the map ψt t. It is clear that 2 ψ is nondecreasing, upper semi-continuous on,, lim inf t t ψt > and ψt < t for all t >. Since Γα i <, we get 2 { a α b α 2 } Λ m Γα Γα 2 One can easily check that 3 Γα Γα 2 H d F x, y, u F x, y, u 2 mx, yψ u u 2. Λ Put X C R,,. Define N : X PX by where N u {h X : there exists v S F,u such that hx, y wx, y for all x, y,, }, wx, y.xα I α 2 γy.yα 2 Γα Γα 2 Iα φx x s α y t α2 vs, tdtds. <.2. Since sup u N u, inf u X sup s N u u s and so N has the approximate endpoint property. Now by using Theorem 2.2, we conclude that the above fractional partial differential inclusion problem has a solution. Acknowledgement Research of the authors was supported by Azarbaijan Shahid Madani University. References Abbas, S., Baleanu, D., Benchohra, M.,Global attractivity for fractional order delay partial integro-differential equations, Adv. Diff. Equ. 22, 22:62. 2 Abbas, S., Benchohra, M., Darboux problem for perturbed partial differential equations of fractional order with finite delay. Nonlinear Anal. Hybrid Syst. 3 29, Abbas, S., Benchohra, M., Fractional order partial hyperbolic differential equations involving Caputo derivative. Stud. Univ. Babes-Bolyai Math. 57 No. 4 22, Abbas, S., Benchohra, M., Partial hyperbolic differential equations with finite delay involving the Caputo fractional derivative. Commun. Math. Anal ,

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